Problem: Write the equation for a parabola with a focus at $(-7,-5)$ and a directrix at $x=-4$. $x=$
Answer: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-7,-5)$, is equal to the distance between $(x,y)$ and the directrix, $x=-4$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-7,-5)$ is $\sqrt{(x+7)^2+(y+5)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $x=-4$ is $\sqrt{(x+4)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(x+4)^2} &= \sqrt{(x+7)^2+(y+5)^2} \\\\ (x+4)^2 &= (x+7)^2+(y+5)^2 \\\\ {x^2}+8x{+16} &= {x^2}{+14x}+49+(y+5)^2\\\\ 8x{-14x}&=(y+5)^2+49{-16} \\\\ -6x&=(y+5)^2+33 \\\\ x&=-\dfrac{(y+5)^2}{6}-\dfrac{11}{2}\end{aligned}$ The answer The equation of our parabola is $x=-\dfrac{(y+5)^2}{6}-\dfrac{11}{2}$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ $y$ $x$ ${(x,y)}$